Diffusion maps embedding and transition matrix analysis of the large-scale flow structure in turbulent Rayleigh--B\'enard convection

By utilizing diffusion maps embedding and transition matrix analysis we investigate sparse temperature measurement time-series data from Rayleigh--B\'enard convection experiments in a cylindrical container of aspect ratio $\Gamma=D/L=0.5$ between its diameter ($D$) and height ($L$). We consider the two cases of a cylinder at rest and rotating around its cylinder axis. We find that the relative amplitude of the large-scale circulation (LSC) and its orientation inside the container at different points in time are associated to prominent geometric features in the embedding space spanned by the two dominant diffusion-maps eigenvectors. From this two-dimensional embedding we can measure azimuthal drift and diffusion rates, as well as coherence times of the LSC. In addition, we can distinguish from the data clearly the single roll state (SRS), when a single roll extends through the whole cell, from the double roll state (DRS), when two counter-rotating rolls are on top of each other. Based on this embedding we also build a transition matrix (a discrete transfer operator), whose eigenvectors and eigenvalues reveal typical time scales for the stability of the SRS and DRS as well as for the azimuthal drift velocity of the flow structures inside the cylinder. Thus, the combination of nonlinear dimension reduction and dynamical systems tools enables to gain insight into turbulent flows without relying on model assumptions

A multidisciplinary approach to the study of shape and motion processing and representation in rats

During my PhD I investigated how shape and motion information are processed by the rat visual system, so as to establish how advanced is the representation of higher-order visual information in this species and, ultimately, to understand to what extent rats can present a valuable alternative to monkeys, as experimental models, in vision studies. Specifically, in my thesis work, I have investigated: 1) The possible visual strategies underlying shape recognition. 2) The ability of rat visual cortical areas to represent motion and shape information. My work contemplated two different, but complementary experimental approaches: psychophysical measurements of the rat\u2019s recognition ability and strategy, and in vivo extracellular recordings in anaesthetized animals passively exposed to various (static and moving) visual stimulation. The first approach implied training the rats to an invariant object recognition task, i.e. to tolerate different ranges of transformations in the object\u2019s appearance, and the application of an mage classification technique known as The Bubbles to reveal the visual strategy the animals were able, under different conditions of stimulus discriminability, to adopt in order to perform the task. The second approach involved electrophysiological exploration of different visual areas in the rat\u2019s cortex, in order to investigate putative functional hierarchies (or streams of processing) in the computation of motion and shape information. Results show, on one hand, that rats are able, under conditions of highly stimulus discriminability, to adopt a shape-based, view-invariant, multi-featural recognition strategy; on the other hand, the functional properties of neurons recorded from different visual areas suggest the presence of a putative shape-based, ventral-like stream of processing in the rat\u2019s visual cortex. The general purpose of my work is and has been the unveiling the neural mechanisms that make object recognition happen, with the goal of eventually 1) be able to relate my findings on rats to those on more visually-advanced species, such as human and non-human primates; and 2) collect enough biological data to support the artificial simulation of visual recognition processes, which still presents an important scientific challenge

On the Geometry of Adiabatic Quantum Mechanics

The adiabatic theorem states that if the Hamiltonian of a quantum system is changed sufficiently slowly, then its instantaneous eigenstates are preserved. In this context, if the original Hamiltonian is restored at the end of the experiment, the phase that the eigenstate has acquired has a purely geometric contribution. This geometric, or Berry, phase is mathematically described using the theory of parallel transport. A generalization of the geometric phase for non-Hermitian Hamiltonians was found by Garrison & Wright, but this does not fit in the standard parallel transport description. The main problem comes from special degeneracies of the Hamiltonian called exceptional points (EPs), around which the energy bands form spiral staircases and so have a non-trivial topology. In this thesis, we introduce a mathematical framework that can cope with EPs and the generalized geometric phase simultaneously. We first treat how Hamiltonians can be studied without using an inner product. This works for all non-degenerate Hamiltonians, which form the first relevant space we study. We then find that the energy bands naturally form a covering space and conclude that the energy permutations due to EPs are described by the monodromy action. Afterwards, we extend this approach to eigenstates, where we find that the geometric phase comes from a natural connection. This provides a rigorous footing for topological geometric phase and the quantum geometric tensor. We finish by demonstrating an equivalent multi-state approach allowing one to express the results using explicit matrices

Multi-view Learning as a Nonparametric Nonlinear Inter-Battery Factor Analysis

Factor analysis aims to determine latent factors, or traits, which summarize a given data set. Inter-battery factor analysis extends this notion to multiple views of the data. In this paper we show how a nonlinear, nonparametric version of these models can be recovered through the Gaussian process latent variable model. This gives us a flexible formalism for multi-view learning where the latent variables can be used both for exploratory purposes and for learning representations that enable efficient inference for ambiguous estimation tasks. Learning is performed in a Bayesian manner through the formulation of a variational compression scheme which gives a rigorous lower bound on the log likelihood. Our Bayesian framework provides strong regularization during training, allowing the structure of the latent space to be determined efficiently and automatically. We demonstrate this by producing the first (to our knowledge) published results of learning from dozens of views, even when data is scarce. We further show experimental results on several different types of multi-view data sets and for different kinds of tasks, including exploratory data analysis, generation, ambiguity modelling through latent priors and classification.Comment: 49 pages including appendi